Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}7x+7y &= 7 \\ x+2y &= -8\end{align*}$
Answer: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-2$ and the bottom equation by $7$ $\begin{align*}-14x-14y &= -14\\ 7x+14y &= -56\end{align*}$ Add the top and bottom equations. $-7x = -70$ Divide both sides by $-7$ and reduce as necessary. $x = 10$ Substitute $10$ for $x$ in the top equation. $7( 10)+7y = 7$ $70+7y = 7$ $7y = -63$ $y = -9$ The solution is $\enspace x = 10, \enspace y = -9$.